{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:CPXQLYACVOK6DD25U3EGKMMQSK","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"18b456f305c512c3e3d9fff5c2efac041a9a70eb6a5e03439afa92eed0de0c8d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2012-06-26T12:54:36Z","title_canon_sha256":"9ca86e3cb43b1349fb5f1b5e333340497e82954a298bc6c000dbf9670009b3c9"},"schema_version":"1.0","source":{"id":"1206.5975","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1206.5975","created_at":"2026-05-18T03:52:37Z"},{"alias_kind":"arxiv_version","alias_value":"1206.5975v1","created_at":"2026-05-18T03:52:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.5975","created_at":"2026-05-18T03:52:37Z"},{"alias_kind":"pith_short_12","alias_value":"CPXQLYACVOK6","created_at":"2026-05-18T12:27:01Z"},{"alias_kind":"pith_short_16","alias_value":"CPXQLYACVOK6DD25","created_at":"2026-05-18T12:27:01Z"},{"alias_kind":"pith_short_8","alias_value":"CPXQLYAC","created_at":"2026-05-18T12:27:01Z"}],"graph_snapshots":[{"event_id":"sha256:b6d5a134c411020d33ff2ff1501d57cf956aff45e1d2fbb89cab16323dbb568f","target":"graph","created_at":"2026-05-18T03:52:37Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"For any small involutive quantaloid Q we define, in terms of symmetric quantaloid-enriched categories, an involutive quantaloid Rel(Q) of Q-sheaves and relations, and a category Sh(Q) of Q-sheaves and functions; the latter is equivalent to the category of symmetric maps in the former. We prove that Rel(Q) is the category of relations in a topos if and only if Q is a modular, locally localic and weakly semi-simple quantaloid; in this case we call Q a Grothendieck quantaloid. It follows that Sh(Q) is a Grothendieck topos whenever Q is a Grothendieck quantaloid. Any locale L is a Grothendieck qua","authors_text":"Hans Heymans, Isar Stubbe","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2012-06-26T12:54:36Z","title":"Grothendieck quantaloids for allegories of enriched categories"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.5975","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:6395e39469a4975bff7079f48d4660aea56a2c910af0b4f02a1a7e90646ab07d","target":"record","created_at":"2026-05-18T03:52:37Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"18b456f305c512c3e3d9fff5c2efac041a9a70eb6a5e03439afa92eed0de0c8d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2012-06-26T12:54:36Z","title_canon_sha256":"9ca86e3cb43b1349fb5f1b5e333340497e82954a298bc6c000dbf9670009b3c9"},"schema_version":"1.0","source":{"id":"1206.5975","kind":"arxiv","version":1}},"canonical_sha256":"13ef05e002ab95e18f5da6c865319092b22715b5dcc9d2eac397c5abece77073","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"13ef05e002ab95e18f5da6c865319092b22715b5dcc9d2eac397c5abece77073","first_computed_at":"2026-05-18T03:52:37.457109Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:52:37.457109Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"pN329/cAnIfdNki+HDNnHQtS8Na6PCY0RJJnnbi1krqf2LqAd3U4gjvs9S8vkDSCGq4n//uSF+Eveogq6zhVCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:52:37.457854Z","signed_message":"canonical_sha256_bytes"},"source_id":"1206.5975","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6395e39469a4975bff7079f48d4660aea56a2c910af0b4f02a1a7e90646ab07d","sha256:b6d5a134c411020d33ff2ff1501d57cf956aff45e1d2fbb89cab16323dbb568f"],"state_sha256":"2a065d718657ce2fa121f71eb4184ea6baecebf6264754361b2cac560ae945f0"}